Solve for x and give the fundamental/general solutions: (Sinx - Cosx)^2 = 1^2Show complete solution and explain the answer.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to expand the binomial `(sin x - cos x)^2`  using the formula `(a-b)^2 = a^2 - 2ab + b^2`  such that:

`(sin x - cos x)^2 = sin^2 x - 2sin x*cos x + cos^2 x`

You need to remember the basic trigonometric formula `sin^2 x+ cos^2 x = 1`  such that:

`(sin x - cos x)^2 = 1- 2sin x*cos x`

You need to substitute `1- 2sin x*cos x`  for `(sin x - cos x)^2`  in your equation such that:

`1- 2sin x*cos x = 1 =gt -2sin x*cos x = 0 =gt sin x*cos x = 0 (since 2!=0)`

`sin x = 0 =gt x = (-1)^n*sin^(-1) 0 + n*pi =gt x = n*pi`

`cos x = 0 =gt x = +-cos^(-1) 0 + 2n*pi`

`x = +-(pi/2) + 2n*pi`

Hence, the general solutions to trigonometric equation are `x = n*pi`  and `x = +-(pi/2) + 2n*pi.`

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